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Journal of Global Optimization

, Volume 47, Issue 3, pp 463–484 | Cite as

Solutions to quadratic minimization problems with box and integer constraints

  • David Yang Gao
  • Ning Ruan
Article

Abstract

This paper presents a canonical duality theory for solving quadratic minimization problems subjected to either box or integer constraints. Results show that under Gao and Strang’s general global optimality condition, these well-known nonconvex and discrete problems can be converted into smooth concave maximization dual problems over closed convex feasible spaces without duality gap, and can be solved by well-developed optimization methods. Both existence and uniqueness of these canonical dual solutions are presented. Based on a second-order canonical dual perturbation, the discrete integer programming problem is equivalent to a continuous unconstrained Lipschitzian optimization problem, which can be solved by certain deterministic technique. Particularly, an analytical solution is obtained under certain condition. A fourth-order canonical dual perturbation algorithm is presented and applications are illustrated. Finally, implication of the canonical duality theory for the popular semi-definite programming method is revealed.

Keywords

Canonical duality theory Quadratic programming Integer programming NP-hard problems Global optimization 

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References

  1. 1.
    Akrotirianakis I.G., Floudas C.A.: Computational experience with a new class of convex underestimators: Box-constrained NLP problems. J. Glob. Optim. 29, 249–264 (2004)CrossRefGoogle Scholar
  2. 2.
    Akrotirianakis I.G., Floudas C.A.: A new class of improved convex underestimators for twice continuously differentiable constrained NLPs. J. Glob. Optim. 30, 367–390 (2004)CrossRefGoogle Scholar
  3. 3.
    Contesse L.: Une caractérisation compléte des minima locaux en programmation quadratique. Numerische Mathematik 34, 315–332 (1980)CrossRefGoogle Scholar
  4. 4.
    Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland (1976)Google Scholar
  5. 5.
    Fang S.C., Gao D.Y., Sheu R.l., Wu S.Y.: Canonical dual approach for solving 0–1 quadratic programming problems. J. Ind. Manag. Optim. 4(1), 125–142 (2008)Google Scholar
  6. 6.
    Fang, S.C., Gao D.Y., Sheu R.l., Xin, W.X.: Global optimization for a class of fractional programming problems, to appear in. J. Glob. Optim. (2008)Google Scholar
  7. 7.
    Floudas C.A.: Deterministic Optimization. Theory, Methods, and Applications. Kluwer, London (2000)Google Scholar
  8. 8.
    Floudas C.A., Akrotirianakis I.G., Caratzoulas S., Meyer C.A., Kallrath J.: Global optimization in the 21th century: Advances and challenges. Comput. Chem. Eng. 29, 1185–1202 (2005)CrossRefGoogle Scholar
  9. 9.
    Floudas C.A., Visweswaran V.: A primal-relaxed dual global optimization approach. J. Optim. Theory Appl. 78(2), 187–225 (1993)CrossRefGoogle Scholar
  10. 10.
    Floudas C.A., Visweswaran V.: Quadratic optimization. In: Horst, R., Pardalos, P.M. (eds) Handbook of Global Optimization, pp. 217–270. Kluwer, Dordrecht/Boston/London (1995)Google Scholar
  11. 11.
    Gao D.Y.: Duality, triality and complementary extremum principles in nonconvex parametric variational problems with applications. IMA J. Appl. Math. 61, 199–235 (1998)CrossRefGoogle Scholar
  12. 12.
    Gao D.Y.: Duality Principles in Nonconvex Systems: Theory, Methods and Applications, 18+454pp. Kluwer, Dordrecht/Boston/London (2000)Google Scholar
  13. 13.
    Gao D.Y.: Canonical dual transformation method and generalized triality theory in nonsmooth global optimization. J. Glob. Optim. 17(1/4), 127–160 (2000)CrossRefGoogle Scholar
  14. 14.
    Gao D.Y.: Perfect duality theory and complete solutions to a class of global optimization problems. Optimization 52(4–5), 467–493 (2003)CrossRefGoogle Scholar
  15. 15.
    Gao D.Y.: Canonical duality theory and solutions to constrained nonconvex quadratic programming. J. Glob. Optim. 29, 377–399 (2004)CrossRefGoogle Scholar
  16. 16.
    Gao D.Y.: Solutions and optimality to box constrained nonconvex minimization problems. J. Ind. Maneg. Optim. 3(2), 293–304 (2007)Google Scholar
  17. 17.
    Gao D.Y.: Canonical duality theory: Unified understanding and generalized solution for global optimization problems. Comp. Chem. Eng. (2009) doi: 10.1016/j.compchemeng.2009.06.009
  18. 18.
    Gao D.Y., Ogden R.W.: Multiple solutions to non-convex variational problems with implications for phase transitions and numerical computation. Quart. J. Mech. Appl. Math. 61(4), 497–522 (2008)CrossRefGoogle Scholar
  19. 19.
    Gao, D.Y., Ruan, N., Sherali, H.D.: Solutions and optimality criteria for nonconvex constrained global optimization problems with connections between canonical and Lagrangian duality, J. Glob. Optim. (2009). doi: 10.1007/s10898-009-9399-x
  20. 20.
    Gao, D.Y., Ruan, N, Walson, L., Tranter, W.H.: Canonical dual approach for solving box and integer constrained minimization problems via a deterministic direct search algorithm. (2009) (in preparation)Google Scholar
  21. 21.
    Gao D.Y., Sherali H.D.: Canonical duality theory: Connections between nonconvex mechanics and global optimization. In: Gao, D.Y., Sherali, H. (eds) Advances in Applied Mathematics and Global Optimization, pp. 257–326. Springer, USA (2009)Google Scholar
  22. 22.
    Gao D.Y., Strang G.: Geometric nonlinearity: Potential energy, complementary energy, and the gap function. Quart. Appl. Math. 47(3), 487–504 (1989)Google Scholar
  23. 23.
    Gao D.Y., Yang W.H.: Multi-duality in minimal surface type problems. Stud. Appl. Math. 95, 127–146 (1995)Google Scholar
  24. 24.
    Goemans M.X., Williamson D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 1115–1145 (1995)CrossRefGoogle Scholar
  25. 25.
    Grippo L., Lucidi S.: A differentiable exact penalty function for bound constrained quadratic programming problems. Optimization 22(4), 557–578 (1991)CrossRefGoogle Scholar
  26. 26.
    Hansen, P., Jaumard, B., Ruiz, M., Xiong, J.: Global minimization of indefinite quadratic functions subject to box constraints. Technical Report G-91–54, GERAD, École Polytechnique, Université McGill, Montreal (1991)Google Scholar
  27. 27.
    Jones D., Perttunen C., Stuckman B.: Lipschitzian optimization without the lipschitz constant. J. Optim. Theory Appl. 79, 157–181 (1993)CrossRefGoogle Scholar
  28. 28.
    Li S.F., Gupta A.: On dual configuration forces. J. Elast. 84, 13–31 (2006)CrossRefGoogle Scholar
  29. 29.
    Murty K.G., Kabadi S.N.: Some NP-hard problems in quadratic and nonlinear programming. Math. Program. 39, 117–129 (1987)CrossRefGoogle Scholar
  30. 30.
    Pardalos P.M., Schnitger G.: Checking local optimality in constrained quadratic and nonlinear programming. Oper. Res. Lett. 7, 33–35 (1988)CrossRefGoogle Scholar
  31. 31.
    Pardalos P., Vavasis S.: Quadratic programming with one negative eigenvalue is NP-hard. J. Glob. Optim. 1, 15–23 (1991)CrossRefGoogle Scholar
  32. 32.
    Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton, NJ (1970)Google Scholar
  33. 33.
    Ruan N., Gao D.Y., Jiao Y. Canonical dual least square method for solving general nonlinear systems of equations. Computational Optimization with Applications (published online: http://www.springerlink.com/content/c6090221p4g41858/). (2008) doi: 10.1007/s10589-008-9222-5
  34. 34.
    Sherali H.D., Tuncbilek C.: A global optimization for polynomial programming problem using a reformulation-linearization technique. J. Glob. Optim. 2, 101–112 (1992)CrossRefGoogle Scholar
  35. 35.
    Sherali H.D., Tuncbilek C.: New reformulation-linearization technique based relaxation for univariate and multivariate polynominal programming problems. Oper. Res. Lett. 21(1), 1–10 (1997)CrossRefGoogle Scholar
  36. 36.
    Todd M.: Semidefinite optimization. Acta Numerica 10, 515–560 (2001)CrossRefGoogle Scholar
  37. 37.
    Wang, Z.B., Fang, S-C., Gao, D.Y., Xing, W.X.: Canonical duality approach to maximum cut problem, to appear (2009)Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2009

Authors and Affiliations

  1. 1.Department of MathematicsVirginia TechBlacksburgUSA

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